(solamente las fichas grandes) 8 Toque un cuadrado amarillo

By definition of the characteristic p a th

rk

defined in equation (4.2.5), the travel tim e Tk can be w ritten as

T* J u[x,t(x)] (43'1)

In order to derive a model for the trav el tim e Tk, we m ake th e three following assum ptions:

1) a t all locations along the stream th ere is a single-valued relationship betw een the w etted cross-sectional area A and discharge Q ;

2) the stream flow a t the upstream location rem ains subcritical; and

3) sources or sinks of w ater along the river are sm all enough so th a t they do not affect the downstream propagation of the river flow.

The first assum ption is only valid for flows th a t are not u n d e r the influence of backw ater effects such as u p stre am of a confluence or weir (Henderson, 1966). To obtain a relationship betw een A and Q u n d e r general conditions requires the solution of the w a ter m om entum equation in the V enant equations. As already m entioned, th is is com putationally d e m a n d in g an d req u ire s in fo rm a tio n t h a t m ay n o t be a v a ila b le . C o n se q u e n tly , th e m o m entum e q u a tio n is o ften re p la c e d by an approxim ation, such as diffusion or kinem atic wave models w ith little loss in accuracy (G unaratm an and Perkins, 1970).

In practice, single-valued relationships betw een A and Q are often invoked from empirical equations. In p articu lar, power law s of th e form

A(x,t) = v /x ) Q(x,t) ^ (4.3.2)

have been used for in-bank flows (Leopold and M addock, 1953; B ates and Pilgrim , 1982) where and r\2 are p a ra m eters u su a lly obtained from stage discharge curves.

The second assum ption is eq u iv alen t to a ssu m in g t h a t th e Froude num ber F defined as the ratio betw een the advective velocity and the w ater

wave celerity is strictly less th an one (Henderson, 1966). In other words, the w a ter w aves trav el faster th a n the w ater parcels. Cases w here F becomes g re a te r th a n one only occur w hen th ere is a sudden change in th e flow regim e due, for exam ple, to an obstacle or a b ru p t m odification of th e channel geom etry.

As for th e th ird assum ption, consider first a fu n ctio n al relatio n sh ip betw een A and Q (invoked in the first assum ption) applied to th e w ater m ass conservation equation (4.2.1). This yields th e relationships

a n d w here dQ dQ , ,, — + w— = w(p - e - a) dt dx (4.3.3) = p - e - d (4.3.4) dt dx w dQ_ d A (4.3.5)

E quations (4.3.3) and (4.3.4) show th a t Q and A are solutions of a first order PDE w ith 'advective' term s w d Q / d x and w d A / d x, respectively. Therefore, th e q u an tity w is equal to the velocity a t which th e flow Q or the w etted cross-sectional area A travel dow nstream . In o th er words, w

is the celerity of a kinem atic w ater wave and equation (4.3.5) is referred to as the Kleitz-Seddon law (Lighthill and W hitham , 1955).

I t is now clear th a t a large source and sink term p - e - d w ill affect the way w ater flow propagates dow nstream . O ur th ird assu m p tio n ensures th a t th is is not th e case so th a t (4.3.3) and (4.3.4) can be reduced to th e ap p ro x im atio n

dt dx dt dx

In order to derive some of the properties of th e w ater wave celerity w

th a t re s u lt from our assum ptions, consider a w a te r wave c h aracteristic curve yxt defined by the equation d x / dt = w (see Figure 4.6). E quation

(4.3.6) yields d Q /d t = cLA/dt - 0 on yxt. Therefore, Q and A, and hence

u = Q /A and w = d Q / d A, rem ain co n stan t along yxt. In addition, because d x / d t = w = const on yxt , th en yxt is a s tra ig h t line. C onsequently, th e value of a k in em atic variable co n stru cted from Q

and/or A a t any point [x,t(x)] e r k can be related to a fixed value in space m easured a t some u p stream location. This featu re re su lts in su b s ta n tia l com putational sim plifications. For exam ple, w ith reference to F igure 4.6, we have on r k

u [x,t(x)J = u° [t(x) -Qx,t(x))] ; and w [x,t(x)] = w° [t (x)-£(x,t(x))]

w here u° is the u p stre am cross-sectional average velocity; w° is th e u p stre am w ater wave celerity; and £ is the w a te r wave trav e l tim e a t

[x,t(x) ] .

t

upstream location downstream location

Figure 4.6 Water wave characteristic paths yLk and yxt and associated wave travel times £k and C(x,t(x)).

We can now w rite (4.3.1) as

r„

dx

or w ith an appropriate change of variable (see Appendix A4)

f

d v j 1 — u°(v) / w °(v) k (4.3.8) w here

C,k is th e travel time of a w ater wave arriving a t L a t tim e k\ u°(v) is th e u p stream cross-sectional average velocity a t tim e v ;

a n d

w°(v) is the upstream w ater wave celerity a t time v.

Note th a t in (4.3.8) the integration is not performed along r k any more. The ratio u°(v)/w°(v) is the upstream Froude num ber denoted F°(v). In the p a rticu la r case where A° and Q° a t the upstream location are related by the power law (4.3.2) w ith p aram eters T f} and rj°2 we have using the Kleitz-Seddon law (4.3.5)

F°(v) = n° (4.3.9)

Therefore F°(v) is constant and the solute travel tim e (4.3.8) becomes

A

F”

(43.10)

To estim ate the wave travel tim e , we use the constancy of wk over the w ater wave characteristic p a th yLk. In other words, we have C,k = L / w k.

Since by definition l / w = d A / d Q and A and Q are related via th e power law (4.3.2), we have

IV t

n,n2Q

w ith Q k b ein g the d o w n stream riv er discharge. T his provides th e following simple expression for the w ater wave travel tim e C,k

If the values of the param eters 77; and tj2 are unavailable or thought to be

inaccurate, equation (4.3.11) can be calibrated using sam ples of £k. These sam ples can be obtained by considering the u p stream and dow nstream flow h y d ro g rap h s a n d m ea su rin g th e tim e elap sed betw een u p s tre a m an d dow nstream flow peaks or troughs. C alib ratio n of (4.3.11) can th e n be perform ed, for exam ple by logarithm ic tran sfo rm atio n which re s u lts in a lin e a r regression model.

E quations (4.3.10) and (4.3.11) yield the solute travel tim e model

\ = (43.12)

w here Qk is m easu red a t the dow nstream location and a and ß a re constants. Model (4.3.12) h as been tested for the River M urray. However, we have observed improved performance if, in stead of being a constant, the Froude num ber F°(v) is assum ed to be related to the flow via the power law

f = aQob (43.13)

The power law (4.3.13) h a s been invoked since F = u / w and often in practice both average cross-sectional velocity u and w ater wave celerity w

are assum ed to be power laws in the stream discharge Q. W ith (4.3.13) the solute travel tim e Tk in (4.3.8) becomes

k - C k f cfv ' l - a Q ° k - T k aQ (v) (43.14)

Note th a t we expect the p aram eter b to be close to zero since F° m ay not depart too m uch from being a constant (such as in (4.3.9)). F u rth erm o re, a

has to be strictly positive and sufficiently sm all to ensure F° < 1, i.e. flows rem ain subcritical. We m ay therefore linearize eq u atio n (4.3.14) in th e neighbourhood of (a,b) = (a^O) where a 0 is sm all and strictly positive. This is done in A ppendix A5, w here we show th a t u pon a p p ro p ria te expansion and linearization we have

w here

&

a1 , a2

ek

< lnQ°>k

is the wave travel tim e given by (4.3.11);

are p a ra m e te rs th a t can be expected to be in th e neighbourhood of i / a and - 6 /a , respectively;

is an error term ; and

is a tim e average defined as

jlnQ°(v)dv (43.16)

The p aram eters a { and a 2 can be estim ated by least squares.

4.3.2 Calibration of the solute travel time m odel Tk

The aim h ere is to p resen t only briefly p e rtin e n t inform ation about salinity d a ta and p a ra m eter e stim ates resu ltin g from model calibration applied to model (4.3.15) since the details are in Dietrich et al (1986b).

50 km

B oundary

B end E u sto n R ed c liffs

x^X'X-x-x?!??

M ild u r a C olignan

< 3 w eirpool

c sa lin ity m ea su re m e n t locations

Q stre a m d ischarge m e a su re m e n t locations

Figure 4.7 Schematic representation of the stretch of the River Murray to which the solute transport model is applied. The letters V and 'Q' denote the locations at which stream salinity and stream discharge measurements are available, respectively. The triangles denote weirpools.

All data used in this study have been supplied by the Murray-Darling Basin Commission. The Commission uses this information as an aid in its management of the flow and the quality of the water in the Basin. Figure 4.7 illustrates the locations where stream flow and stream salinity data are measured on a daily basis.

Table 4.1 provides the frequency of daily salinity recordings obtained at Boundary Bend, Euston, Redcliffs and Mildura for the periods October 1967 to December 1976 and January 1977 to December 1983. The distinction between both periods was made for the purpose of calibrating the model over the first time period and then testing the model over the second time period. Solute travel time data were abstracted by matching well-defined upstream peaks (and troughs) in salinity profiles over time with associated and well-defined downstream peaks (and troughs).

Table 4.1 Frequency of daily salinity recordings obtained at Boundary Bend, Euston, Redcliffs and Mildura for the periods October 1967 to December 1976 and January 1977 to December 1983.

Measurement Frequency of daily salinity recordings (%) location

October 1967-December 1976 January 1977-December 1983

Boundary Bend 65.3 32.7

Euston 65.3 66.4

Redcliffs 66.6 66.4

M ildura 65.6 64.7

In Table 4.2, we provide the sample size for solute travel time data used in the calibration of the travel time model (4.3.14) for the period from October 1967 to December 1976.

Table 4.3 contains the results of calibrating the solute travel time model (4.3.15) for the three reaches with the wave travel time given by

Ck = .0758 LQ~k ' 67 (43.17)

where Qk is the downstream river flow in megalitres/day; and L is the length of the reach in kilometers. The wave travel time expression (4.3.17)

w a s o b tain ed from (4.3.11) by considering u p stre a m an d d o w n stream h isto ric a l flow h y d ro g rap h s an d m easu rin g th e tim e elap sed betw een u p stre am and dow nstream flow peaks or troughs. C alib ratio n w as th en perform ed by logarithm ic transform ation and lin ear regression.

Table 4.2 Sample size for travel time data used in the calibration of the solute travel time model (4.3.15) for the period from October 1967 to December 1976 .

R each Sam ple

size

Percentage of num ber of sam p les to to ta l days in sam pling period (%)

B oundary Bend to E uston 49 1.45

E uston to Redcliffs 60 1.78

E uston to M ildura 64 1.89

Table 4.3 Values for parameters in the solute travel time model (4.3.15), their standard deviations and the R? statistic, as obtained via least squares.

Reach p a ra m e te r p a ra m e te r a 2 m e a n sta n d a rd deviation m e a n s ta n d a rd deviation R 2 Boundary Bend to Euston 18.38 .89 -1.68 .095 .911 Euston to Redcliffs 11.48 .38 -1.02 .04 .953 Euston to M ild u ra 11.56 .63 -1.01 .07 .874

In Table 4.3 the coefficient of determination R 2 is given by

R 2 = 1 - ---

A

k = /

where K is the number of solute travel time samples z*k; r* is the mean of the and is the estimated solute travel time obtained from (4.3.15).

Note again that validation results are not be presented here since they are given in Dietrich et al (1986b).

4.4 MODEL FOR THE INTEGRAL J k

In document Capítulo 2: Etiologías del Daño Cerebral (página 163-171)